Basic Analysis I — Real Analysis · Problem 7.1.6

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

Let

(X, d_X)

and

(Y, d_Y)

be metric spaces. a) Show that

(X \times Y, d)

with

d\bigl((x_1,y_1),(x_2,y_2)\bigr) \coloneq d_X(x_1,x_2) + d_Y(y_1,y_2)

is a metric space. b) Show that

(X \times Y, d)

with

d\bigl((x_1,y_1),(x_2,y_2)\bigr) \coloneq \max\{d_X(x_1,x_2), d_Y(y_1,y_2)\}

is a metric space.